Finite Morse Index Solutions Research Articles

Finite Morse index solutions of a nonlinear Schrödinger equation in half-space with nonlinear boundary value conditions

Finite Morse index solutions of a nonlinear Schrödinger equation in half-space with nonlinear boundary value conditions

Uniform boundedness for finite Morse index solutions to supercritical semilinear elliptic equations

AbstractWe consider finite Morse index solutions to semilinear elliptic questions, and we investigate their smoothness. It is well‐known that: For , there exist Morse index 1 solutions whose norm goes to infinity. For , uniform boundedness holds in the subcritical case for power‐type nonlinearities, while for critical nonlinearities the boundedness of the Morse index does not prevent blow‐up in . In this paper, we investigate the case of general supercritical nonlinearities inside convex domains, and we prove an interior a priori bound for finite Morse index solution in the sharp dimensional range . As a corollary, we obtain uniform bounds for finite Morse index solutions to the Gelfand problem constructed via the continuity method.

Classification of finite Morse index solutions to the polyharmonic Hénon equation

Classification of finite Morse index solutions to the polyharmonic Hénon equation

Classification of Finite Morse Index Solutions for Robin Boundary Value Problems

Classification of Finite Morse Index Solutions for Robin Boundary Value Problems

Liouville‐type theorem for finite Morse index solutions to the Choquard equation involving Δλ‐Laplacian

In this paper, we are concerned with the following Choquard‐type equation: where , and is a sub‐elliptic operator of the form Under some general conditions of , we prove that the equation has no nontrivial solution which is stable outside a compact set in the subcritical case , where is the homogeneous dimension of associated to . In addition, for the critical case , we shall show that any solution of the equation above, which is stable outside a compact set, has finite energy. More precisely, we show that

On stable and finite Morse index solutions to quasilinear Schrödinger equations

On stable and finite Morse index solutions to quasilinear Schrödinger equations

Classification and Liouville-type theorems for semilinear elliptic equations in unbounded domains

We classify stable and finite Morse index solutions to general semilinear elliptic equations posed in Euclidean space of dimension at most 10, or in some unbounded domains.

Finite Morse Index Solutions of the Fractional Henon–Lane–Emden Equation with Hardy Potential

In this paper, we study the fractional Henon–Lane–Emden equation associated with Hardy potential \[ (-\Delta)^{s} u - \gamma |x|^{-2s} u = |x|^a |u|^{p-1} u \quad \textrm{in $\mathbb{R}^{n}$}. \] Extending the celebrated result of [14], we obtain a classification result on finite Morse index solutions to the fractional elliptic equation above with Hardy potential. In particular, a critical exponent $p$ of Joseph–Lundgren type is derived in the supercritical case studying a Liouville type result for the $s$-harmonic extension problem.

Classification of finite Morse index solutions to the elliptic sine-Gordon equation in the plane

The elliptic sine-Gordon equation is a semilinear elliptic equation with a special double well potential. It has a family of explicit multiple-end solutions. We show that all finite Morse index solutions belong to this family. It will also be proved that these solutions are nondegenerate, in the sense that the corresponding linearized operators have no nontrivial bounded kernel. Finally, we prove that the Morse index of 2n -end solutions is equal to n(n-1)/2 .

Stable and finite Morse index solutions of a nonlinear Schrödinger system

Stable and finite Morse index solutions of a nonlinear Schrödinger system

On stable and finite Morse index solutions of the nonlocal Hénon-Gelfand–Liouville equation

We consider the nonlocal Hénon-Gelfand–Liouville problem $$\begin{aligned} (-\Delta )^s u = |x|^a e^u\quad \mathrm {in}\quad \mathbb {R}^n, \end{aligned}$$for every \(s\in (0,1)\), \(a>0\) and \(n>2s\). We prove a monotonicity formula for solutions of the above equation using rescaling arguments. We apply this formula together with blow-down analysis arguments and technical integral estimates to establish non-existence of finite Morse index solutions when $$\begin{aligned} \dfrac{\Gamma (\frac{n}{2})\Gamma (s)}{\Gamma (\frac{n-2s}{2})}\left( s+\frac{a}{2}\right) > \dfrac{\Gamma ^2(\frac{n+2s}{4})}{\Gamma ^2(\frac{n-2s}{4})}. \end{aligned}$$

Recent progress on stable and finite Morse index solutions of semilinear elliptic equations

<p style='text-indent:20px;'>We discuss some recent results (mostly from the last decade) on stable and finite Morse index solutions of semilinear elliptic equations, where Norman Dancer has made many important contributions. Some open questions in this direction are also discussed.</p>

On stable and finite Morse index solutions of the fractional Toda system

We develop a monotonicity formula for solutions of the fractional Toda system(−Δ)sfα=e−(fα+1−fα)−e−(fα−fα−1)inRn, when 0<s<1, α=1,⋯,Q, f0=−∞, fQ+1=∞ and Q≥2 is the number of equations in this system. We then apply this formula, technical integral estimates, classification of stable homogeneous solutions, and blow-down analysis arguments to establish Liouville type theorems for finite Morse index (and stable) solutions of the above system when n>2s andΓ(n2)Γ(1+s)Γ(n−2s2)Q(Q−1)2>Γ2(n+2s4)Γ2(n−2s4). Here, Γ is the Gamma function. When Q=2, the above equation is the classical (fractional) Gelfand-Liouville equation.

Some Liouville theorems for Hénon type equations in half-space with nonlinear boundary value conditions and finite Morse indices

In this paper, we are concerned with Liouville-type theorems for Henon type equations in half-space with nonlinear boundary value conditions. We prove Liouville-type theorems for solutions belonging to one of the following classes: stable solutions and finite Morse index solutions (whether positive or sign-changing). Our proof is based on a combination of the integral estimates, the Pohozaev-type identity and the monotonicity formula of solutions and blowing down sequence.

On finite Morse index solutions of higher order fractional elliptic equations

<p style='text-indent:20px;'>We consider sign-changing solutions of the equation <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ (-{\Delta})^s u+\lambda u = |u|^{p-1}u \; \mbox{in}\; \mathbb R^n, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M1">\begin{document}$ n\geq 1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ \lambda&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ p&gt;1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ 1&lt;s\leq2 $\end{document}</tex-math></inline-formula>. The main goal of this work is to analyze the influence of the linear term <inline-formula><tex-math id="M5">\begin{document}$ \lambda u $\end{document}</tex-math></inline-formula>, in order to classify stable solutions possibly unbounded and sign-changing. We prove Liouville type theorems for stable solutions or solutions which are stable outside a compact set of <inline-formula><tex-math id="M6">\begin{document}$ \mathbb R^n $\end{document}</tex-math></inline-formula>. We first derive a monotonicity formula for our equation. After that, we provide integral estimate from stability which combined with Pohozaev-type identity to obtain nonexistence results in the subcritical case with the restrictive condition <inline-formula><tex-math id="M7">\begin{document}$ |u|_{L^{\infty}( \mathbb R^n)}^{p-1}&lt; \frac{\lambda (p+1) }{2} $\end{document}</tex-math></inline-formula>. The supercritical case needs more involved analysis, motivated by the monotonicity formula, we then reduce the nonexistence of nontrivial entire solutions which are stable outside a compact set of <inline-formula><tex-math id="M8">\begin{document}$ \mathbb R^n $\end{document}</tex-math></inline-formula>. Through this approach we give a complete classification of stable solutions <b>for all <inline-formula><tex-math id="M9">\begin{document}$ p&gt;1 $\end{document}</tex-math></inline-formula></b>. Moreover, for the case <inline-formula><tex-math id="M10">\begin{document}$ 0&lt;s\leq1 $\end{document}</tex-math></inline-formula>, finite Morse index solutions are classified in [<xref ref-type="bibr" rid="b19">19</xref>,<xref ref-type="bibr" rid="b25">25</xref>].

Finite Morse index solutions of the H\xe9non Lane\u2013Emden equation

In this paper, we are concerned with Liouville-type theorems of the Hénon Lane–Emden triharmonic equations in whole space. We prove Liouville-type theorems for solutions belonging to one of the following classes: stable solutions and finite Morse index solutions (whether positive or sign-changing). Our proof is based on a combination of the Pohozaev-type identity, monotonicity formula of solutions and a blowing down sequence.

Stable and finite Morse index solutions of Toda system

We prove a Liouville type theorem and a partial regularity result for stable and finite Morse index solutions of Toda system. The main tools are integral estimates, a monotonicity formula and blowing down analysis. For such systems, they may be split into two sub-systems during the blowing down process. Some ε-regularity theorems are developed to deal with this problem.

Monotonicity formulas for coupled elliptic gradient systems with applications

AbstractConsider the following coupled elliptic system of equations$$\begin{array}{} \displaystyle (-{\it\Delta})^s u_i = (u^2_1+\cdots+u^2_m)^{\frac{p-1}{2}} u_i \quad \text{in} ~~ \mathbb{R}^n , \end{array}$$where 0 &lt;s≤ 2,p&gt; 1,m≥ 1,u=$\begin{array}{} \displaystyle (u_i)_{i=1}^m \end{array}$andui: ℝn→ ℝ. The qualitative behavior of solutions of the above system has been studied from various perspectives in the literature including the free boundary problems and the classification of solutions. For the case of local scalar equation, that is whenm= 1 ands= 1, Gidas and Spruck in [26] and later Caffarelli, Gidas and Spruck in [6] provided the classification of solutions for Sobolev sub-critical and critical exponents. More recently, for the case of local system of equations that is whenm≥ 1 ands= 1 a similar classification result is given by Druet, Hebey and Vétois in [17] and references therein. In this paper, we derive monotonicity formulae for entire solutions of the above local, whens= 1, 2, and nonlocal, when 0 &lt;s&lt; 1 and 1 &lt;s&lt; 2, system. These monotonicity formulae are of great interests due to the fact that a counterpart of the celebrated monotonicity formula of Alt-Caffarelli-Friedman [1] seems to be challenging to derive for system of equations. Then, we apply these formulae to give a classification of finite Morse index solutions. In the end, we provide an open problem in regards to monotonicity formulae for Lane-Emden systems.

On the classification of solutions of cosmic strings equation

In this paper, we prove some Liouville-type theorems for weak stable or finite Morse index solutions to the following equation: $$\begin{aligned} \Delta u+e^u+|x|^\alpha e^{\beta u}=0\quad \mathrm {in}\quad \mathbb {R}^N, \end{aligned}$$ which arises from the study of selfgravitating cosmic strings for a massive W-boson model coupled with Einstein’s equation.

Finite Morse Index Implies Finite Ends

AbstractWe prove that finite Morse index solutions to the Allen‐Cahn equation in ℝ2 have finitely many ends and linear energy growth. The main tool is a curvature decay estimate on level sets of these finite Morse index solutions, which in turn is reduced to a problem on the uniform second‐order regularity of clustering interfaces for the singularly perturbed Allen‐Cahn equation. Using an indirect blowup technique, in the spirit of the classical Colding‐Minicozzi theory in minimal surfaces, we show that the obstruction to the uniform second‐order regularity of clustering interfaces in ℝn is associated to the existence of nontrivial entire solutions to a (finite or infinite) Toda system in ℝn–1. For finite Morse index solutions in ℝ2, we show that this obstruction does not exist by using information on stable solutions of the Toda system. © 2019 Wiley Periodicals, Inc.